| L(s) = 1 | + 25·5-s − 108·7-s − 8·11-s + 162·13-s + 714·17-s + 532·19-s − 4.58e3·23-s + 625·25-s − 938·29-s + 8.36e3·31-s − 2.70e3·35-s + 1.09e3·37-s + 1.12e4·41-s + 7.69e3·43-s − 1.36e4·47-s − 5.14e3·49-s − 1.90e4·53-s − 200·55-s − 1.89e4·59-s − 1.97e3·61-s + 4.05e3·65-s − 4.42e4·67-s − 5.97e4·71-s + 5.69e4·73-s + 864·77-s + 1.51e4·79-s − 2.19e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.833·7-s − 0.0199·11-s + 0.265·13-s + 0.599·17-s + 0.338·19-s − 1.80·23-s + 1/5·25-s − 0.207·29-s + 1.56·31-s − 0.372·35-s + 0.130·37-s + 1.04·41-s + 0.634·43-s − 0.900·47-s − 0.306·49-s − 0.931·53-s − 0.00891·55-s − 0.708·59-s − 0.0680·61-s + 0.118·65-s − 1.20·67-s − 1.40·71-s + 1.25·73-s + 0.0166·77-s + 0.272·79-s − 0.350·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 108 T + p^{5} T^{2} \) |
| 11 | \( 1 + 8 T + p^{5} T^{2} \) |
| 13 | \( 1 - 162 T + p^{5} T^{2} \) |
| 17 | \( 1 - 42 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 28 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4584 T + p^{5} T^{2} \) |
| 29 | \( 1 + 938 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 - 1090 T + p^{5} T^{2} \) |
| 41 | \( 1 - 11238 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7692 T + p^{5} T^{2} \) |
| 47 | \( 1 + 13640 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19050 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18936 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1978 T + p^{5} T^{2} \) |
| 67 | \( 1 + 44212 T + p^{5} T^{2} \) |
| 71 | \( 1 + 59744 T + p^{5} T^{2} \) |
| 73 | \( 1 - 56994 T + p^{5} T^{2} \) |
| 79 | \( 1 - 15128 T + p^{5} T^{2} \) |
| 83 | \( 1 + 21996 T + p^{5} T^{2} \) |
| 89 | \( 1 + 14066 T + p^{5} T^{2} \) |
| 97 | \( 1 - 75938 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.478226908027263014278184978812, −8.328425939803989061197293649684, −7.52709434742087623865314076750, −6.31667637436776613224940603645, −5.91212787273480141023114530287, −4.63514162846776284970224048118, −3.53789781884061580991090194177, −2.56321959033502310158946192163, −1.29705061490019575620931133911, 0,
1.29705061490019575620931133911, 2.56321959033502310158946192163, 3.53789781884061580991090194177, 4.63514162846776284970224048118, 5.91212787273480141023114530287, 6.31667637436776613224940603645, 7.52709434742087623865314076750, 8.328425939803989061197293649684, 9.478226908027263014278184978812
